Mathed PDF Presentit Fractals

7/28/2019 Mathed PDF Presentit Fractals

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Some

preliminarywords


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Some

preliminarywords

All underlined words arelinks.

I have placed a framearound every image in thepresentation that wasmade by or taken bysomeone else. Thecaptions underneath theframed images link to the

web pages they camefrom.


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Some

preliminarywords

All underlined words arelinks.

I have placed a framearound every image in thepresentation that wasmade by or taken bysomeone else. Thecaptions underneath theframed images link to the

web pages they camefrom.

Who isPeitgen?
http://sprott.physics.wisc.edu/fractals/chaos/http://sprott.physics.wisc.edu/fractals/chaos/http://sprott.physics.wisc.edu/fractals/chaos/http://sprott.physics.wisc.edu/fractals/chaos/


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Fractals:

a Symmetry approach
http://www.math.umass.edu/~mconnors/fractal/fractal.htmlhttp://en.wikipedia.org/wiki/Symmetryhttp://en.wikipedia.org/wiki/Symmetryhttp://www.math.umass.edu/~mconnors/fractal/fractal.html


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First discussed will be threecommon types of symmetry:

Reflectional (Line or Mirror)

Rotational (N-fold)

Translational


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First discussed will be threecommon types of symmetry:

Reflectional (Line or Mirror)

Rotational (N-fold)

Translational

and then: the Magnification(Dilatational a.k.a. Dilational)

symmetry of fractals.
http://dict.die.net/dilatation/http://dict.die.net/dilatation/


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Reflectional (aka Line or Mirror) Symmetry

A shape exhibits reflectional symmetry if the shape can bebisected by a line L, one half of the shape removed, and themissing piece replaced by a reflection of the remaining pieceacross L, then the resulting combination is (approximately) thesame as the original.1

From An Intuitive Notion of Line Symmetry
http://classes.yale.edu/fractals/IntroToFrac/SelfSim/Symmetry/ReflSymmetry.htmlhttp://mathworld.wolfram.com/Bisection.htmlhttp://classes.yale.edu/fractals/IntroToFrac/SelfSim/Symmetry/ReflSymmetry.htmlhttp://regentsprep.org/Regents/math/symmetry/Lsymmet.htmhttp://classes.yale.edu/fractals/IntroToFrac/SelfSim/Symmetry/ReflSymmetry.htmlhttp://mathworld.wolfram.com/Bisection.htmlhttp://classes.yale.edu/fractals/IntroToFrac/SelfSim/Symmetry/ReflSymmetry.htmlhttp://regentsprep.org/Regents/math/symmetry/Lsymmet.htm


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Reflectional (aka Line or Mirror) Symmetry

reflectional symmetryA shape exhibits if the shape can bebisected by a line L, one half of the shape removed, and themissing piece replaced by a reflection of the remaining pieceacross L, then the resulting combination is (approximately) thesame as the original.1

From An Intuitive Notion of Line Symmetry

In simpler words, if you

can fold it over and itmatches up, it has

reflectional symmetry.

This leaf, and the butterfly

caterpi llar sitt ing on it, areroughly symmetric. So are

human faces. Line

symmetry and mirror

symmetry are terms that

mean the same thing.
http://classes.yale.edu/fractals/IntroToFrac/SelfSim/Symmetry/ReflSymmetry.htmlhttp://mathworld.wolfram.com/Bisection.htmlhttp://classes.yale.edu/fractals/IntroToFrac/SelfSim/Symmetry/ReflSymmetry.htmlhttp://regentsprep.org/Regents/math/symmetry/Lsymmet.htmhttp://classes.yale.edu/fractals/IntroToFrac/SelfSim/Symmetry/ReflSymmetry.htmlhttp://mathworld.wolfram.com/Bisection.htmlhttp://classes.yale.edu/fractals/IntroToFrac/SelfSim/Symmetry/ReflSymmetry.htmlhttp://regentsprep.org/Regents/math/symmetry/Lsymmet.htm


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The butterfly and the

children have lines ofreflection symmetry

where one sidemirrors the other.

Taken at the same time atthe Desert Botanical Gardens

Butterfly Pavilion, the littlebutterfly is a Painted Lady(Vanessa, cardui). Its host

plant is (Thistles, cirsium).
http://desertbotanical.org/index.aspx?pageID=554http://desertbotanical.org/index.aspx?pageID=554http://desertbotanical.org/index.aspx?pageID=554http://desertbotanical.org/index.aspx?pageID=554http://www.fs.fed.us/r4/htnf/resources/wildflowers/thistles.shtmlhttp://www.fs.fed.us/r4/htnf/resources/wildflowers/thistles.shtmlhttp://desertbotanical.org/index.aspx?pageID=554http://desertbotanical.org/index.aspx?pageID=554http://desertbotanical.org/index.aspx?pageID=554


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Here is a link to a PowerPointpresentation created by Mrs. Gamache

using the collection of web pages by theAdrian Bruce and students of 6B.

This site lets you create your ownsymmetry patterns! Choose your typeand color, then start moving the mouseand clicking.

Both images have

a curved line ofsymmetry at the

edge of the water.
http://www.adrianbruce.com/Symmetry/power/LineSymmetry.ppthttp://www.adrianbruce.com/Symmetry/power/LineSymmetry.ppthttp://www.adrianbruce.com/Symmetry/power/LineSymmetry.ppthttp://www.adrianbruce.com/Symmetry/power/LineSymmetry.ppthttp://www.scienceu.com/geometry/handson/kali/index.cgi?group=w632http://www.scienceu.com/geometry/handson/kali/index.cgi?group=w632http://www.scienceu.com/geometry/handson/kali/index.cgi?group=w632http://www.scienceu.com/geometry/handson/kali/index.cgi?group=w632http://www.scienceu.com/geometry/handson/kali/index.cgi?group=w632http://www.scienceu.com/geometry/handson/kali/index.cgi?group=w632http://www.scienceu.com/geometry/handson/kali/index.cgi?group=w632http://www.adrianbruce.com/Symmetry/power/LineSymmetry.ppthttp://www.adrianbruce.com/Symmetry/power/LineSymmetry.ppthttp://www.adrianbruce.com/Symmetry/power/LineSymmetry.ppt


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These blooms have 5 fold rotationalThese blooms have 5 fold rotational


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These blooms have 5-fold rotationalsymmetry. They can be turned

5 times to leave the figure unchanged

before starting over again.




These blooms have 5 fold rotationalThese blooms have 5-fold rotational


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The butterfly is aSpicebush Swallowtail(Papillo, troilus).

The butterfly is aSpicebush Swallowtail(Papillo, troilus).

A pentagon also has 5-fold symmetry.A pentagon also has 5-fold symmetry.


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An example of 4-fold rotational symmetry,a property shared by the square.An example of 4-fold rotational symmetry,a property shared by the square.


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The tiny blooms have 4-fold

symmetry. Question: doesthe spherical bloom they sit

on have n-fold symmetry?

This flower has21-fold rotational

symmetry.

5-fold or 6-foldsymmetry here


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T l ti l S t


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The bricks in the image have

translational symmetry.

The bricks in the image have

translational symmetry.

Translational Symmetry

A shape exhibits translational symmetry if displacementin some direction - horizontal or vertical, for example -returns the shape to (approximately) its original

configuration.3

Also, the image of the bricks willhave translation symmetry when

sliding, provided there is norotation during the move.

Also, the image of the bricks willhave translation symmetry when

sliding, provided there is norotation during the move.

Orientation must be preservedwhile translating.

Orientation must be preservedwhile translating.

M ifi ti (Dil t ti l) S t
http://classes.yale.edu/fractals/IntroToFrac/SelfSim/Symmetry/TransSymmetry.htmlhttp://classes.yale.edu/fractals/IntroToFrac/SelfSim/Symmetry/TransSymmetry.htmlhttp://classes.yale.edu/fractals/IntroToFrac/SelfSim/Symmetry/TransSymmetry.htmlhttp://classes.yale.edu/fractals/IntroToFrac/SelfSim/Symmetry/TransSymmetry.html


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Magnification (Dilatational) Symmetry

symmetry under magnificationLess familiar is :zooming in on an object leaves the shape

approximately unaltered.4

Zooming in on a fractal objectleaves the shape

approximately unaltered.

Fractals exhibit magnification symmetry.
http://classes.yale.edu/fractals/IntroToFrac/SelfSim/Symmetry/Symmetry.htmlhttp://classes.yale.edu/fractals/IntroToFrac/SelfSim/Symmetry/Symmetry.htmlhttp://classes.yale.edu/fractals/IntroToFrac/SelfSim/Symmetry/Symmetry.htmlhttp://classes.yale.edu/fractals/IntroToFrac/SelfSim/Symmetry/Symmetry.html


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Natural Fractals

Multifractals

Chaos

Natural fractals have a limited number ofstages of growth, and the growthbetween stages shows variation. They

have connections to Multifractals andChaos theory.
http://www.umanitoba.ca/faculties/science/botany/labs/ecology/fractals/applications.htmlhttp://www.matpack.de/Info/Mathematics/Multifractals.htmlhttp://www.khwarzimic.org/activities/chaos-intro.htmlhttp://www.khwarzimic.org/activities/chaos-intro.htmlhttp://www.matpack.de/Info/Mathematics/Multifractals.htmlhttp://www.umanitoba.ca/faculties/science/botany/labs/ecology/fractals/applications.html


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Fractal geometry was designedto handle shapes that appear

complicated, but with complexityarranged in some hierarchical

fashion. So at a minimum,fractals must have some

substructure.

(Michael Frame, Yale University)

This is a Sweet Acacia (Acacia, smallii) tree. Its
http://www.enature.com/flashcard/show_flash_card.asp?recordNumber=TS0252http://www.enature.com/flashcard/show_flash_card.asp?recordNumber=TS0252


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This is a Sweet Acacia (Acacia, smallii) tree. Itsunbloomed flower appears to be a sphere madeup ofsmaller-scale spheres, but a closer look

reveals the little buds to be cylindrical.

Outer foliage (leaves and petals) onOuter foliage (leaves and petals) on
http://www.enature.com/flashcard/show_flash_card.asp?recordNumber=TS0252http://math.bu.edu/DYSYS/chaos-game/node5.htmlhttp://math.bu.edu/DYSYS/chaos-game/node5.htmlhttp://www.enature.com/flashcard/show_flash_card.asp?recordNumber=TS0252


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Ou e o age (ea es a d pe as) oplants are usually terminal organs,

and are non-reproductive. (There are

exceptions, though, like the entire fernfamily.) Root Gorelick of the ASU

biology department explains:

Ou e o age (ea es a d pe as) oplants are usually terminal organs,

and are non-reproductive. (There are

exceptions, though, like the entire fernfamily.) Root Gorelick of the ASU

biology department explains:

Leaves are terminal

organs, hence don'treproduce miniature

copies of themselves as

do stems, roots, andmany reproductive

structures. Therefore, I

expect leaves to be leastfractal of these organs.

(Root Gorelick)

Leaves are terminal

organs, hence don'treproduce miniature

copies of themselves as

do stems, roots, andmany reproductive

structures. Therefore, I

expect leaves to be least

fractal of these organs.

(Root Gorelick)

The bloom of the SweetA i f h
http://www.biology-online.org/dictionary/terminalhttp://math.bu.edu/DYSYS/dysys.htmlhttp://www.biology-online.org/dictionary/terminalhttp://math.bu.edu/DYSYS/dysys.htmlhttp://math.bu.edu/DYSYS/dysys.htmlhttp://math.bu.edu/DYSYS/dysys.htmlhttp://www.biology-online.org/dictionary/terminalhttp://www.biology-online.org/dictionary/terminal


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Image from Paul Bourkes

Acacia tree from theprevious slide isreproductive, but that is

not enough to be fractal.Having a fractalsubstructure requiresthe same or a highly

similar shape betweena minimum of 3 stagesof growth (there existssome disagreement on

this, Im going with Yale,a good source).

The entire fernfamily reveals self-

similarity:

successive stages

of growth thatclosely resemble

earlier stages. Self-Similaritypage.

Butterfly is aButterfly is a
http://astronomy.swin.edu.au/~pbourke/fractals/selfsimilar/index.htmlhttp://astronomy.swin.edu.au/~pbourke/fractals/selfsimilar/index.html


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11

22

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There are severalstages of scaling

not visible in thisimage, see nextimage where the

top of the plant isvisible.

There are severalstages of scaling

not visible in thisimage, see nextimage where the

top of the plant isvisible.

Butterfly is aZebra Swallowtail

(Eurytides, marcellus)

Butterfly is aZebra Swallowtail

(Eurytides, marcellus)

The butterfly is aThe butterfly is aCl dl Gi S l h


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The scaled

branching extendsupward throughoutthe plant. A small

branch, if magnified,would look like a

larger branch.

The scaled

branching extendsupward throughoutthe plant. A small

branch, if magnified,would look like a

larger branch.

Cloudless Giant Sulpher(Phoebis, sennae)

Cloudless Giant Sulpher(Phoebis, sennae)


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Also notice how entiresections of the plant

resemble each other ondifferent scales. Thestructure of smaller

sections dictates theshape of larger sections.

Also notice how entiresections of the plant

resemble each other ondifferent scales. Thestructure of smaller

sections dictates theshape of larger sections.


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Fractal branching isFractal branching is


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captured in shadowbelow. From thisview, again noticehow the partsresemble thewhole.

captured in shadowbelow. From thisview, again noticehow the partsresemble thewhole.

A Painted Lady is present!A Painted Lady is present!


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The plant in the

previous slidesresembles thiscomputer-

generated binaryfractal tree.

Image by Don West
http://classes.yale.edu/http://classes.yale.edu/fractals/FracTrees/welcome.htmlhttp://faculty.plattsburgh.edu/don.west/trees/http://faculty.plattsburgh.edu/don.west/trees/http://classes.yale.edu/http://classes.yale.edu/fractals/FracTrees/welcome.html


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With fractals, the structure behind small sections

dictates overall shape.

We saw empirical verification of this in the previousexample. We saw that the bigger shapes wereaggregations of the smaller shapes that made them

up. This is also true of clouds, mountains, oceanwaves, lightning, and many other aspects of nature.An ocean wave is made up of a lot of little waves,

which are in turn made up of yet smaller waves. Thisis why fractal equations tend to be simple.

Tremendous complexity can result fromiterating

simple patterns.
http://dict.die.net/empirical/http://dict.die.net/aggregation/http://mathforum.org/library/drmath/view/54531.htmlhttp://mathforum.org/library/drmath/view/54531.htmlhttp://dict.die.net/aggregation/http://dict.die.net/empirical/


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Image courtesy of Paul Bourke


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Beware of complacency! Many aspects ofnature are fractal, while others are not. Ofthose aspects that have an embedded

fractal structure, their fractal aspect onlydescribes properties of shape andcomplexity. Read this Word of Caution

from Nonlinear Geoscience: Fractals.The randomness referred to in theirstatement is given consideration in

Multifractal theory, which has ties toChaos theory and Nonlinear Dynamics.

Geometric Fractals
http://journal-ci.csse.monash.edu.au/ci/vol06/jelinek/jelinek.htmlhttp://ems.gphys.unc.edu/nonlinear/fractals/geometry.htmlhttp://www.physics.mcgill.ca/~gang/multifrac/clouds/clouds.htmhttp://www.geocities.com/Athens/6398/chaos.htmhttp://amath.colorado.edu/faculty/jdm/faq-%5b2%5d.htmlhttp://amath.colorado.edu/faculty/jdm/faq-%5b2%5d.htmlhttp://www.geocities.com/Athens/6398/chaos.htmhttp://www.physics.mcgill.ca/~gang/multifrac/clouds/clouds.htmhttp://ems.gphys.unc.edu/nonlinear/fractals/geometry.htmlhttp://journal-ci.csse.monash.edu.au/ci/vol06/jelinek/jelinek.html


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Geometric Fractals

Geometric fractals might be compared to objects/systems

in a vacuumin physics, in that abnormalities arenonexistent in them. They are, as their name suggests,geometric constructs, perfect (Ideal) systems with nointernal deviations or potential changes from outside

influences (apart from human error in constructing them).Geometric fractals have no randomness, and noconnections to Chaos theory.

Im not including Complex fractals in this category such asthe Mandelbrot Set, J ulia Sets, or any fractal that lies in the

complex plane. Complex fractals are highlighted later.

The Sierpinski Tetrahedron
http://csep10.phys.utk.edu/guidry/violence/lightspeed.htmlhttp://csep10.phys.utk.edu/guidry/violence/lightspeed.html


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pFractal type: Geometric

The number of tetrahedra is increasing in powers of 4The edge-length of the tetrahedra is decreasing in powers of

The volume of Sierpinskis tetrahedron is decreasing in powers of

Image created using MathCad by Byrge Birkeland of Agder University College, Kristiansand, Norway

To consider this fractal, it is important to know something about
http://home.hia.no/~byrgeb/imageslinks.htmhttp://home.hia.no/~byrgeb/imageslinks.htmhttp://en.wikipedia.org/wiki/Tetrahedron


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a tetrahedron.

To consider this fractal, it is important to know something abouth d
http://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Tetrahedron


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a tetrahedron.

- Start with an equilateral triangle.

To consider this fractal, it is important to know something aboutt t h d
http://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Tetrahedron


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a tetrahedron.

- Start with an equilateral triangle.- Divide it into 4 equilateral triangles by marking the midpointsof all three sides and drawing lines to connect the midpoints.
http://en.wikipedia.org/wiki/Tetrahedronhttp://www.mathwords.com/e/equilateral_triangle.htmhttp://www.mathwords.com/e/equilateral_triangle.htmhttp://en.wikipedia.org/wiki/Tetrahedron


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To construct a stage-1:

Reduce it by afactor of 1/2

Start with a regular tetrahedron.

It is called the stage-0 in theSierpinski tetrahedron fractalfamily.

Replicate (4 are needed). Thetetrahedra are kept transparent onthis slide to reinforce that these

are tetrahedra and not triangles.

Rebuild the 4 stage-0s into a
http://www.emints.org/ethemes/resources/S00001378.shtmlhttp://www.mathwords.com/t/tetrahedron.htmhttp://www.emints.org/ethemes/resources/S00001378.shtmlhttp://www.mathwords.com/t/tetrahedron.htm


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Rebuild the 4 stage-0 s into astage-1 Sierpinski tetrahedron.

The line is a handy frame ofreference for construction.

Rebuild the 4 stage-0s into a


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Rebuild the 4 stage 0 s into astage-1 Sierpinski tetrahedron.

The line is a handy frame ofreference for construction.

Now repeat this process againand again.

Revisiting the earlier image, notice that each tetrahedron isl d b 4 t t h d i th t t


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replaced by 4 tetrahedra in the next stage.

Determine the stage by counting the number ofsizes of openings,

the stage-1 has one size of opening, the stage-2 two sizes ofopenings, etc

Revisiting the earlier image, notice that each tetrahedron isl d b 4 t t h d i th t t


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Revisiting the earlier image, notice that each tetrahedron isreplaced by 4 tetrahedra in the next stage


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The Sierpinski Triangle: grows in Powers of 3
http://math.rice.edu/~lanius/fractals/http://math.rice.edu/~lanius/fractals/


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Reduceby again

Replicate& Rebuild

Reduceby

Replicate& Rebuild

Notice how each trianglebecomes three triangles

in the next stage.

Geometric fractals are typically filling or emptying something, whether it is

length, surface area, or volume. The key points are that dimension is: 1)changing, and 2) generally fractional.

The stage can be

determined by thenumber of differentsizes of openings.

123

4

With this fractal, it is

surface area instead ofvolume that is decreasingat each stage.

The face of a Sierpinski

tetrahedron is a same-stage Sierpinski triangle.
http://math.bu.edu/DYSYS/chaos-game/node6.htmlhttp://math.bu.edu/DYSYS/chaos-game/node6.html


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A quick word about the Chaos Game. It is a neat game thatuses a randomprocess to create the Sierpinski Triangle You
http://math.bu.edu/DYSYS/applets/chaos-game.htmlhttp://math.bu.edu/DYSYS/applets/chaos-game.html


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uses a random process to create the Sierpinski Triangle. You

can play it at the above link.

The Sierpinski triangle and other Geometric fractals can becreated using random processes like the Chaos Game; however,as stated in this link, the fractals created by such games are notchaotic. Geometric fractals typically map into the Cartesian

and have no connections with mathematical Chaos.plane

A quick word about the Chaos Game. It is a neat game thatuses a randomprocess to create the Sierpinski Triangle You
http://www.geocities.com/ResearchTriangle/System/8956/Fractal/intro.htmhttp://mathworld.wolfram.com/CartesianPlane.htmlhttp://mathworld.wolfram.com/CartesianPlane.htmlhttp://mathworld.wolfram.com/CartesianPlane.htmlhttp://mathworld.wolfram.com/CartesianPlane.htmlhttp://mathworld.wolfram.com/CartesianPlane.htmlhttp://mathworld.wolfram.com/CartesianPlane.htmlhttp://mathworld.wolfram.com/CartesianPlane.htmlhttp://mathworld.wolfram.com/CartesianPlane.htmlhttp://www.geocities.com/ResearchTriangle/System/8956/Fractal/intro.htmhttp://math.bu.edu/DYSYS/applets/chaos-game.htmlhttp://math.bu.edu/DYSYS/applets/chaos-game.html


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uses a random process to create the Sierpinski Triangle. You

can play it at the above link.

The Sierpinski triangle and other Geometric fractals can becreated using random processes like the Chaos Game; however,as stated in this link, the fractals created by such games are notchaotic. Geometric fractals typically map into the Cartesian

plane and have no connections with mathematical Chaos.

The Chaos Game was named more than 20 years ago whenthere was no real definition of chaos. One qualification: I donot include the Mandelbrot Set and J ulia SetsComplex fractals

that lie in the complex plane and have significant connections toChaosin the Geometric fractals category.

It is easier to talk about fractal
http://www.geocities.com/ResearchTriangle/System/8956/Fractal/intro.htmhttp://mathworld.wolfram.com/CartesianPlane.htmlhttp://mathworld.wolfram.com/CartesianPlane.htmlhttp://mathworld.wolfram.com/CartesianPlane.htmlhttp://mathworld.wolfram.com/CartesianPlane.htmlhttp://mathworld.wolfram.com/CartesianPlane.htmlhttp://mathworld.wolfram.com/ComplexPlane.htmlhttp://mathworld.wolfram.com/ComplexPlane.htmlhttp://mathworld.wolfram.com/CartesianPlane.htmlhttp://mathworld.wolfram.com/CartesianPlane.htmlhttp://mathworld.wolfram.com/CartesianPlane.htmlhttp://www.geocities.com/ResearchTriangle/System/8956/Fractal/intro.htmhttp://www.cs.cornell.edu/Courses/cs312/2000fa/handouts/fractals.htmlhttp://www.cs.cornell.edu/Courses/cs312/2000fa/handouts/fractals.htmlhttp://www.cs.cornell.edu/Courses/cs312/2000fa/handouts/fractals.htmlhttp://www.cs.cornell.edu/Courses/cs312/2000fa/handouts/fractals.htmlhttp://www.cs.cornell.edu/Courses/cs312/2000fa/handouts/fractals.html


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dimension in Geometric fractalsbecause the property is more exact

in them, even though it applies toall types of fractal objects and

systems.

Even though most fractals havenon-integer dimension, there are

exceptions:

For exactly self-similar shapes made of N copies,each scaled by a factor of r, the dimension is
http://www.cs.cornell.edu/Courses/cs312/2000fa/handouts/fractals.htmlhttp://www.cs.cornell.edu/Courses/cs312/2000fa/handouts/fractals.htmlhttp://www.cs.cornell.edu/Courses/cs312/2000fa/handouts/fractals.htmlhttp://www.cs.cornell.edu/Courses/cs312/2000fa/handouts/fractals.htmlhttp://www.cs.cornell.edu/Courses/cs312/2000fa/handouts/fractals.html


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each scaled by a factor of r, the dimension is

Log(N)/Log(1/r)

For exactly self-similar shapes made of N copies,each scaled by a factor of r, the dimension is


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each scaled by a factor of r, the dimension is

Log(N)/Log(1/r)

The Sierpinski tetrahedron is made of N = 4 copies,each scaled by a factor of r = 1/2, so its dimension is

Log(4)/Log(2) = 2
http://classes.yale.edu/fractals/Labs/SierpTetraLab/SierpTetraLab.htmlhttp://classes.yale.edu/fractals/Labs/SierpTetraLab/SierpTetraLab.html


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Contrast this with the Sierpinski triangle, made ofN = 3 copies each scaled by a factor of r =


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N = 3 copies, each scaled by a factor of r = .

Its dimension is

Log(3)/Log(2) ~= 1.58496..

Contrast this with the Sierpinski triangle, made ofN = 3 copies each scaled by a factor of r =


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N 3 copies, each scaled by a factor of r .

Its dimension is

Log(3)/Log(2) ~= 1.58496..

The Sierpinski triangle has fractional dimension,

more typical of fractals.

Contrast this with the Sierpinski triangle, made ofN = 3 copies, each scaled by a factor of r = .
http://math.rice.edu/~lanius/fractals/dim.htmlhttp://math.rice.edu/~lanius/fractals/dim.html


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N 3 copies, each scaled by a factor of r .

Its dimension is

Log(3)/Log(2) ~= 1.58496..

The Sierpinski triangle has fractional dimension,

more typical of fractals.

The exact answer is Log(3)/Log(2). Theapproximate answer (often more useful in realworld applications) is the decimal approximation1.58496
http://math.rice.edu/~lanius/fractals/dim.htmlhttp://www.trottermath.net/probsolv/toplug.htmlhttp://www.trottermath.net/probsolv/toplug.htmlhttp://math.rice.edu/~lanius/fractals/dim.html


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Self-similarity: this is a bigidea, and it only trulyapplies to geometric

fractals; however, it isused as a concept to talk

about all types of fractals.

Something is self-similar when every
http://math.rice.edu/~lanius/fractals/selfsim.htmlhttp://math.rice.edu/~lanius/fractals/selfsim.html


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Something is self-similar when every

little part looks exactly like the whole.The only place this can really happen

is in a perfect (Ideal) system at infinity.Only geometric fractals have a chance

of arriving at this state; however, inorder to speak about fractals

generally, one must embrace theconcept of self-similarity in a broad

way.

hCh

(All categories listed on this slidelink to relevant websites.)

(All categories listed on this slidelink to relevant websites.)
http://classes.yale.edu/fractals/Chaos/ChaosIntro/ChaosIntro.htmlhttp://classes.yale.edu/fractals/Chaos/ChaosIntro/ChaosIntro.htmlhttp://classes.yale.edu/fractals/Chaos/ChaosIntro/ChaosIntro.html


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ChaosChaos

MultifractalsMultifractals

Random FractalsRandom Fractals

Mandelbrot SetMandelbrot Set

J ulia SetsJ ulia Sets

Image courtesy of J im Muth

Complex FractalsComplex FractalsMandelbrot discusses fractalsMandelbrot discusses fractals

ChaosChaos
http://classes.yale.edu/fractals/Chaos/ChaosIntro/ChaosIntro.htmlhttp://classes.yale.edu/fractals/Chaos/ChaosIntro/ChaosIntro.htmlhttp://classes.yale.edu/fractals/MultiFractals/welcome.htmlhttp://classes.yale.edu/fractals/MultiFractals/welcome.htmlhttp://classes.yale.edu/fractals/RandFrac/welcome.htmlhttp://classes.yale.edu/fractals/RandFrac/welcome.htmlhttp://www.ddewey.net/mandelbrot/noad.htmlhttp://www.ddewey.net/mandelbrot/noad.htmlhttp://www.ibiblio.org/e-notes/MSet/Period.htmhttp://www.ibiblio.org/e-notes/MSet/Period.htmhttp://www.geom.uiuc.edu/~zietlow/defp1.htmlhttp://www.geom.uiuc.edu/~zietlow/defp1.htmlhttp://www.yale.edu/opa/v31.n20/story6.htmlhttp://www.yale.edu/opa/v31.n20/story6.htmlhttp://classes.yale.edu/fractals/Chaos/ChaosIntro/ChaosIntro.htmlhttp://classes.yale.edu/fractals/Chaos/ChaosIntro/ChaosIntro.htmlhttp://www.geom.uiuc.edu/~zietlow/defp1.htmlhttp://www.yale.edu/opa/v31.n20/story6.htmlhttp://classes.yale.edu/fractals/Chaos/ChaosIntro/ChaosIntro.htmlhttp://classes.yale.edu/fractals/Chaos/ChaosIntro/ChaosIntro.htmlhttp://classes.yale.edu/fractals/MultiFractals/welcome.htmlhttp://www.ibiblio.org/e-notes/MSet/Period.htmhttp://www.ddewey.net/mandelbrot/noad.htmlhttp://classes.yale.edu/fractals/RandFrac/welcome.html


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This image is scale-independent. It hasno frame of reference to indicate the

This image is scale-independent. It hasno frame of reference to indicate thei f th l d h i l


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size of the clouds, such as an airplane,or the horizon.

size of the clouds, such as an airplane,or the horizon.

Magnification symmetryrequires a frame ofreference to determine size

because zooming inreveals approximately thesame shape(s).

Magnification symmetryrequires a frame ofreference to determine size

because zooming inreveals approximately thesame shape(s).

Taken by Ralph Kresge.Click inside frame to visitNational Weather Service

(NOAA) photo library

Fractals are scale independent. Recall that small partsaggregate to dominate overall shape.

Fractals are scale independent. Recall that small partsaggregate to dominate overall shape.

Within a fractal system, the smallestscale is present in multitudinous


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scale is present in multitudinous

numbers. The medium scale has asignificant presence, with acomparative handful of giants.

We see examples of this in bugsand galaxies, also in stars within

galaxies. The small areproliferate while the huge are fewand far between.
http://www.umanitoba.ca/faculties/science/botany/labs/ecology/fractals/applications.htmlhttp://classes.yale.edu/fractals/Panorama/Astronomy/Galaxies/Galaxies.htmlhttp://classes.yale.edu/fractals/Panorama/Astronomy/Galaxies/Galaxies.htmlhttp://www.umanitoba.ca/faculties/science/botany/labs/ecology/fractals/applications.html


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Butterfly Wing Branching PatternsButterfly Wing Branching Patterns


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A closer view would reveal fractal branching in the veins of the wings of this ZebraSwallowtail. Almost all branching in nature is fractal. Leaf veins are an another instance

of fractal branching. And rivers. And ourcirculatory system. And lightning.

From the Bugbios website:the Caligo genus
http://www.stat.rice.edu/~riedi/UCDavisHemoglobin/fractal.htmlhttp://www.photovault.com/Link/Orders/Flora/Leaves/OFLVolume01.htmlhttp://classes.yale.edu/fractals/Panorama/Nature/Rivers/Rivers.htmlhttp://classes.yale.edu/fractals/Panorama/Biology/Physiology/Physiology.htmlhttp://www.museum.vic.gov.au/scidiscovery/lightning/shapes.asphttp://www.museum.vic.gov.au/scidiscovery/lightning/shapes.asphttp://classes.yale.edu/fractals/Panorama/Biology/Physiology/Physiology.htmlhttp://classes.yale.edu/fractals/Panorama/Nature/Rivers/Rivers.htmlhttp://www.photovault.com/Link/Orders/Flora/Leaves/OFLVolume01.htmlhttp://www.stat.rice.edu/~riedi/UCDavisHemoglobin/fractal.htmlhttp://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/index.html


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Images from Bugbios

by Dexter Sear

From the Bugbios website:the Caligo genus
http://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/caligo.htmlhttp://www.insects.org/class/patterns/caligo.htmlhttp://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/index.html


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Images from Bugbios

by Dexter Sear

This almost looks like a computer-generated fractalimage, but it isnt. It is a collage of Caligo butterfly wings.Notice the scaled rippling pattern in isolated wingsections. It is a fairly safe conjecture that these ripplesmight have fractal properties. This isnt to suggest that allcolorful butterfly wing patterns would reveal fractal scaling.
http://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/caligo.htmlhttp://www.insects.org/class/patterns/caligo.htmlhttp://www.insects.org/class/patterns/index.html


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http://astronomy.swin.edu.au/~pbourke/fractals/diaxialplane/index.html


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Length analog, the Cantor Set

Area analog, the Sierpinski Carpet

Volume analog, the Menger Sponge

Images courtesy ofPaul Bourke
http://en.wikipedia.org/wiki/Cantor_sethttp://ecademy.agnesscott.edu/~lriddle/ifs/carpet/carpet.htmhttp://planetmath.org/encyclopedia/MengerSponge.htmlhttp://astronomy.swin.edu.au/~pbourke/fractals/http://astronomy.swin.edu.au/~pbourke/fractals/carpet/index.htmlhttp://astronomy.swin.edu.au/~pbourke/fractals/apollony/http://astronomy.swin.edu.au/~pbourke/fractals/diaxialplane/index.htmlhttp://ecademy.agnesscott.edu/~lriddle/ifs/carpet/carpet.htmhttp://astronomy.swin.edu.au/~pbourke/fractals/http://planetmath.org/encyclopedia/MengerSponge.htmlhttp://en.wikipedia.org/wiki/Cantor_set


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Using the same pattern,

20 stage-1s can be puttogether to form a stage-2 with 20x20 = 202 = 400cubes. A cube is beingemptied of its volume.Watch how quickly thisexponential growth getsout of control.


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At each stage, the edge-length of the last cube isreduced by 1/3, andreplicated 20 times. Sothe Menger sponge has

fractal dimension:

log (20)/log (3) =approximately 2.7268


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How far can this go?

As far as you want it to.

There is no reason to stop here.


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This is a stage-6. It is made up of

26 = 64 million cubes.

There is no uncertainty about the

way it wil l grow or what it wil l look

like after any number of stages ofgrowth.

p

Image courtesy ofPaul Bourke.

Fractals Across the DisciplinesFractals Across the Disciplines
http://astronomy.swin.edu.au/~pbourkehttp://astronomy.swin.edu.au/~pbourke


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a handful of topics from the Yale Fractal Geometryweb pageA Panorama of Fractals and Their Uses:

Art & Nature Music

Architecture Nature & Fractals

Astronomy PhysiologyFinance Poetry

HistoryPsychology

Industry Social Sciences

Literature

(The categories all link to their respective pages.)
http://classes.yale.edu/fractals/http://classes.yale.edu/fractals/Panorama/http://classes.yale.edu/fractals/Panorama/Art/ArtAndNature/ArtAndNature.htmlhttp://classes.yale.edu/fractals/Panorama/Music/Mus/Music.htmlhttp://classes.yale.edu/fractals/Panorama/Architecture/Arch/Arch.htmlhttp://classes.yale.edu/fractals/Panorama/Nature/NAtFracGallery/NatFracGallery.htmlhttp://classes.yale.edu/fractals/Panorama/Astronomy/Galaxies/Galaxies.htmlhttp://classes.yale.edu/fractals/Panorama/Biology/Physiology/Physiology.htmlhttp://classes.yale.edu/fractals/Panorama/SocialSciences/FinRisk/FinRisk.htmlhttp://classes.yale.edu/fractals/Panorama/Literature/Poetry/Poetry.htmlhttp://classes.yale.edu/fractals/Panorama/SocialSciences/History/History.htmlhttp://classes.yale.edu/fractals/Panorama/SocialSciences/Psychology/Psychology.htmlhttp://classes.yale.edu/fractals/Panorama/ManuFractals/ManuFract/ManuFractals.htmlhttp://classes.yale.edu/fractals/Panorama/SocialSciences/SocSci/SocSci.htmlhttp://classes.yale.edu/fractals/Panorama/Literature/Lit/Literature.htmlhttp://classes.yale.edu/fractals/Panorama/SocialSciences/SocSci/SocSci.htmlhttp://classes.yale.edu/fractals/Panorama/SocialSciences/Psychology/Psychology.htmlhttp://classes.yale.edu/fractals/Panorama/Literature/Poetry/Poetry.htmlhttp://classes.yale.edu/fractals/Panorama/Biology/Physiology/Physiology.htmlhttp://classes.yale.edu/fractals/Panorama/Nature/NAtFracGallery/NatFracGallery.htmlhttp://classes.yale.edu/fractals/Panorama/Music/Mus/Music.htmlhttp://classes.yale.edu/fractals/Panorama/ManuFractals/ManuFract/ManuFractals.htmlhttp://classes.yale.edu/fractals/Panorama/Literature/Lit/Literature.htmlhttp://classes.yale.edu/fractals/Panorama/Astronomy/Galaxies/Galaxies.htmlhttp://classes.yale.edu/fractals/Panorama/SocialSciences/FinRisk/FinRisk.htmlhttp://classes.yale.edu/fractals/Panorama/SocialSciences/History/History.htmlhttp://classes.yale.edu/fractals/Panorama/Architecture/Arch/Arch.htmlhttp://classes.yale.edu/fractals/Panorama/Art/ArtAndNature/ArtAndNature.htmlhttp://classes.yale.edu/fractals/Panorama/http://classes.yale.edu/fractals/


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Clint Sprott

made thisimage from an

IFS written byPeitgen.
http://sprott.physics.wisc.edu/sprott.htmhttp://www.cs.wpi.edu/~matt/courses/cs563/talks/cbyrd/pres1.htmlhttp://www.cs.wpi.edu/~matt/courses/cs563/talks/cbyrd/pres1.htmlhttp://sprott.physics.wisc.edu/sprott.htmhttp://sprott.physics.wisc.edu/fractals/chaos/


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Who isPeitgen?

Clint Sprott


http://sprott.physics.wisc.edu/fractals/chaos/http://sprott.physics.wisc.edu/fractals/chaos/http://sprott.physics.wisc.edu/sprott.htmhttp://www.cs.wpi.edu/~matt/courses/cs563/talks/cbyrd/pres1.htmlhttp://sprott.physics.wisc.edu/fractals/chaos/http://sprott.physics.wisc.edu/fractals/chaos/http://www.cs.wpi.edu/~matt/courses/cs563/talks/cbyrd/pres1.htmlhttp://sprott.physics.wisc.edu/sprott.htmhttp://sprott.physics.wisc.edu/fractals/chaos/


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Who isPeitgen?

Clint SprottHeinz-Otto Peitgenuses fractalresearch in the arena of medicine toassist surgeons in identifying andoperating on tumors. MeVis dealswith medical research. CeViseducates teachers about fractal

geometry and math/science/art/musicconnections. FAU runs a sisterprogram of CeVis, where I learnedabout fractals, directed by Peitgen

and



Richard F.Voss.
http://sprott.physics.wisc.edu/fractals/chaos/http://sprott.physics.wisc.edu/fractals/chaos/http://sprott.physics.wisc.edu/sprott.htmhttp://weblog.science.fau.edu/info/?p=69http://www.cs.wpi.edu/~matt/courses/cs563/talks/cbyrd/pres1.htmlhttp://web.gc.cuny.edu/sciart/0102/fractals.htmlhttp://web.gc.cuny.edu/sciart/0102/fractals.htmlhttp://weblog.science.fau.edu/info/?p=69http://sprott.physics.wisc.edu/fractals/chaos/http://sprott.physics.wisc.edu/fractals/chaos/http://www.cs.wpi.edu/~matt/courses/cs563/talks/cbyrd/pres1.htmlhttp://sprott.physics.wisc.edu/sprott.htm


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Ready to generatesome fractals using

the computer?

Visit Peitgen andVoss fractal

games website.

These gamesrequire J ava .
http://www.math.fau.edu/MLogan/Pattern_Exploration/MainDirections.htmlhttp://www.math.fau.edu/MLogan/Pattern_Exploration/MainDirections.htmlhttp://www.math.fau.edu/MLogan/Pattern_Exploration/MainDirections.htmlhttp://www.math.fau.edu/MLogan/Pattern_Exploration/MainDirections.htmlhttp://www.math.fau.edu/MLogan/Pattern_Exploration/MainDirections.html


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To build real-worldmodels of

Sierpinskistetrahedron:

Visit mySierpinski Build

page for 5th gradeand under. Older

grades will preferYales methodusing envelopes.
http://www.public.asu.edu/~starlite/sierpinskibuild.htmlhttp://classes.yale.edu/fractals/Labs/SierpTetraLab/SierpTetraLab.htmlhttp://classes.yale.edu/fractals/Labs/SierpTetraLab/SierpTetraLab.htmlhttp://www.public.asu.edu/~starlite/sierpinskibuild.html


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To build real-worldmodels of the

Menger Sponge:

The Business Card cubemethod should work well

for grades 7th and up. Forgrades 4th through 6th, itmight be best to instead

build paper cubes usingtape and tabs, and then

tape the cubes together to

construct the Sponge.
http://www.nedbatchelder.com/text/cardcube.htmlhttp://www.nedbatchelder.com/text/cardcube.html


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Recapping the main fractal theme addressed in thispresentation:


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Fractals operate under a Symmetry ofMagnification (called Dilatation or Dilation inliterature). Different types of fractals share acommon ground of parts that are similar to thewhole. Even though self-similar substructure

must technically be present all the way to infinityfor something to be called fractal, the concept offractility is loosened to apply to forms (esp.

natural) with only a handful of levels ofsubstructure present.

The simplification of complexity leading to useful results that wehave been looking at is not unique to the field of fractals, it is atheme that runs throughout mathematics, with varying methods

f i lifi i


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of simplification.


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of simplification.

Mathematics is about making clean simplified concepts out

of things that we notice in the world around us. In the world(staying with fractals as an example), when it is applicable wemake a clean concept by assuming the existence of self-similarityinfinite levels of substructurewhen there are only

a few, pushing beyond reality. (Priscilla Greenwood,Statistician and Mathematical Biologist at Arizona State

University)


f i lifi ti


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of simplification.

Mathematics is about making clean simplified concepts out

of things that we notice in the world around us. In the world(staying with fractals as an example), when it is applicable wemake a clean concept by assuming the existence of self-similarityinfinite levels of substructurewhen there are only

a few, pushing beyond reality. (Priscilla Greenwood,Statistician and Mathematical Biologist at Arizona State

University)

Math works because these simplified systems work.Mathematicians could well be called The Great Simplifiers, but

that is another presentation, and another day.

Mathed PDF Presentit Fractals

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Transcript of Mathed PDF Presentit Fractals